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Mirrors > Home > MPE Home > Th. List > Mathboxes > mexval | Structured version Visualization version GIF version |
Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mexval.k | ⊢ 𝐾 = (mTC‘𝑇) |
mexval.e | ⊢ 𝐸 = (mEx‘𝑇) |
mexval.r | ⊢ 𝑅 = (mREx‘𝑇) |
Ref | Expression |
---|---|
mexval | ⊢ 𝐸 = (𝐾 × 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mexval.e | . 2 ⊢ 𝐸 = (mEx‘𝑇) | |
2 | fveq2 6446 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇)) | |
3 | mexval.k | . . . . . 6 ⊢ 𝐾 = (mTC‘𝑇) | |
4 | 2, 3 | syl6eqr 2832 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾) |
5 | fveq2 6446 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇)) | |
6 | mexval.r | . . . . . 6 ⊢ 𝑅 = (mREx‘𝑇) | |
7 | 5, 6 | syl6eqr 2832 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅) |
8 | 4, 7 | xpeq12d 5386 | . . . 4 ⊢ (𝑡 = 𝑇 → ((mTC‘𝑡) × (mREx‘𝑡)) = (𝐾 × 𝑅)) |
9 | df-mex 31983 | . . . 4 ⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡))) | |
10 | fvex 6459 | . . . . 5 ⊢ (mTC‘𝑡) ∈ V | |
11 | fvex 6459 | . . . . 5 ⊢ (mREx‘𝑡) ∈ V | |
12 | 10, 11 | xpex 7240 | . . . 4 ⊢ ((mTC‘𝑡) × (mREx‘𝑡)) ∈ V |
13 | 8, 9, 12 | fvmpt3i 6547 | . . 3 ⊢ (𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅)) |
14 | xp0 5806 | . . . . 5 ⊢ (𝐾 × ∅) = ∅ | |
15 | 14 | eqcomi 2787 | . . . 4 ⊢ ∅ = (𝐾 × ∅) |
16 | fvprc 6439 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅) | |
17 | fvprc 6439 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mREx‘𝑇) = ∅) | |
18 | 6, 17 | syl5eq 2826 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅) |
19 | 18 | xpeq2d 5385 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (𝐾 × 𝑅) = (𝐾 × ∅)) |
20 | 15, 16, 19 | 3eqtr4a 2840 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅)) |
21 | 13, 20 | pm2.61i 177 | . 2 ⊢ (mEx‘𝑇) = (𝐾 × 𝑅) |
22 | 1, 21 | eqtri 2802 | 1 ⊢ 𝐸 = (𝐾 × 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ∅c0 4141 × cxp 5353 ‘cfv 6135 mTCcmtc 31960 mRExcmrex 31962 mExcmex 31963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-mex 31983 |
This theorem is referenced by: mexval2 31999 msubff 32026 msubco 32027 msubff1 32052 mvhf 32054 msubvrs 32056 |
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